MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brdom Structured version   Visualization version   GIF version

Theorem brdom 8121
Description: Dominance relation. (Contributed by NM, 15-Jun-1998.)
Hypothesis
Ref Expression
bren.1 𝐵 ∈ V
Assertion
Ref Expression
brdom (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem brdom
StepHypRef Expression
1 bren.1 . 2 𝐵 ∈ V
2 brdomg 8119 . 2 (𝐵 ∈ V → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
31, 2ax-mp 5 1 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wex 1841  wcel 2127  Vcvv 3328   class class class wbr 4792  1-1wf1 6034  cdom 8107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-xp 5260  df-rel 5261  df-cnv 5262  df-dm 5264  df-rn 5265  df-fn 6040  df-f 6041  df-f1 6042  df-dom 8111
This theorem is referenced by:  domen  8122  domtr  8162  sbthlem10  8232  1sdom  8316  ac10ct  9018  domtriomlem  9427  2ndcdisj  21432  birthdaylem3  24850
  Copyright terms: Public domain W3C validator